Write the equation for a parabola with a focus at $(-2,5)$ and a directrix at $y=3$. $y=$
Answer: The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(-2,5)$, is equal to the distance between $(x,y)$ and the directrix, $y=3$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(-2,5)$ is $\sqrt{(x+2)^2+(y-5)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $y=3$ is $\sqrt{(y-3)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(y-3)^2} &= \sqrt{(x+2)^2+(y-5)^2} \\\\ (y-3)^2 &= (x+2)^2+(y-5)^2 \\\\ {y^2}-6y{+9} &= (x+2)^2{+y^2}{-10y}+25 \\\\ -6y{+10y}&=(x+2)^2+25{-9} \\\\ 4y&=(x+2)^2+16 \\\\ y&=\dfrac{(x+2)^2}{4}+4\end{aligned}$ The answer The equation of our parabola is $y=\dfrac{(x+2)^2}{4}+4$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ $y$ $x$ ${(x,y)}$